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Power of a Quotient Property

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Power of a Quotient Property

Study Guide

Key Definition

For any real numbers $a$, $b$, and $c$ (as long as $b \neq 0$), the power of a quotient property states: $(\frac{a}{b})^c = \frac{a^c}{b^c}$

Important Notes

This property is similar to the power of a product property.
Ensure the base denominator $b$ is not zero.
Applies to any real numbers $a$ and $c$ (with $b \neq 0$); negative bases with non-integer exponents are not real.
Helps simplify expressions with exponents.
Commonly used in algebra and calculus.

Mathematical Notation

$\frac{a}{b}$ is a fraction with numerator $a$ and denominator $b$

Remember to use proper notation when solving problems

Why It Works

When you raise a quotient to an exponent, you multiply the quotient by itself $c$ times: $(a/b)^c = (a/b)\times(a/b)\times\dots\times(a/b)$. Multiplying the numerators and denominators separately yields $a^c/b^c$.

Remember

The power of a quotient property is essential for simplifying expressions with division and exponents. Keep in mind $(\frac{a}{b})^c = \frac{a^c}{b^c}$.

Quick Reference

Property:$(\frac{a}{b})^c = \frac{a^c}{b^c}$

Understanding Power of a Quotient Property

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with $(\frac{a}{b})^c = \frac{a^c}{b^c}$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Simplify $\left(\frac{2}{3}\right)^2$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have $\left(\frac{1}{2}\right)^3$ of a chocolate bar. What fraction of the bar is that?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\left(\frac{a}{b}\right)^2 = \frac{9}{16}$, find possible positive values for $a$ and $b$.

Challenge Quiz

Single Choice Quiz

Advanced

Evaluate $\left(\frac{x^2}{y^3}\right)^4$

Recap

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Review key concepts and takeaways